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Metamath Proof Explorer

Theorem dalawlem9

Description: Lemma for dalaw . Special case to eliminate the requirement -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) in dalawlem1 . (Contributed by NM, 6-Oct-2012)

Ref Expression
Hypotheses dalawlem.l 
|- .<_ = ( le ` K )
dalawlem.j 
|- .\/ = ( join ` K )
dalawlem.m 
|- ./\ = ( meet ` K )
dalawlem.a 
|- A = ( Atoms ` K )
Assertion dalawlem9 
|- ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )

Proof

Step Hyp Ref Expression
1 dalawlem.l 
 |-  .<_ = ( le ` K )
2 dalawlem.j 
 |-  .\/ = ( join ` K )
3 dalawlem.m 
 |-  ./\ = ( meet ` K )
4 dalawlem.a 
 |-  A = ( Atoms ` K )
5 simp11 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> K e. HL )
6 5 hllatd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> K e. Lat )
7 simp22 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> Q e. A )
8 simp32 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> T e. A )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) e. ( Base ` K ) )
11 5 7 8 10 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( Q .\/ T ) e. ( Base ` K ) )
12 simp21 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> P e. A )
13 simp31 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> S e. A )
14 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
15 5 12 13 14 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( P .\/ S ) e. ( Base ` K ) )
16 9 3 latmcom 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = ( ( P .\/ S ) ./\ ( Q .\/ T ) ) )
17 6 11 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = ( ( P .\/ S ) ./\ ( Q .\/ T ) ) )
18 simp12 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) )
19 simp23 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> R e. A )
20 2 4 hlatjcom 
 |-  ( ( K e. HL /\ R e. A /\ P e. A ) -> ( R .\/ P ) = ( P .\/ R ) )
21 5 19 12 20 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( R .\/ P ) = ( P .\/ R ) )
22 18 21 breqtrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ R ) )
23 17 22 eqbrtrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( P .\/ R ) )
24 simp13 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) )
25 17 24 eqbrtrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( R .\/ U ) )
26 simp33 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> U e. A )
27 1 2 3 4 dalawlem8 
 |-  ( ( ( K e. HL /\ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( P .\/ R ) /\ ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( R .\/ U ) ) /\ ( Q e. A /\ P e. A /\ R e. A ) /\ ( T e. A /\ S e. A /\ U e. A ) ) -> ( ( Q .\/ P ) ./\ ( T .\/ S ) ) .<_ ( ( ( P .\/ R ) ./\ ( S .\/ U ) ) .\/ ( ( R .\/ Q ) ./\ ( U .\/ T ) ) ) )
28 5 23 25 7 12 19 8 13 26 27 syl333anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ P ) ./\ ( T .\/ S ) ) .<_ ( ( ( P .\/ R ) ./\ ( S .\/ U ) ) .\/ ( ( R .\/ Q ) ./\ ( U .\/ T ) ) ) )
29 2 4 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
30 5 12 7 29 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
31 2 4 hlatjcom 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) = ( T .\/ S ) )
32 5 13 8 31 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( S .\/ T ) = ( T .\/ S ) )
33 30 32 oveq12d 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) = ( ( Q .\/ P ) ./\ ( T .\/ S ) ) )
34 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) )
35 5 7 19 34 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
36 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) )
37 5 8 26 36 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( T .\/ U ) e. ( Base ` K ) )
38 9 3 latmcl 
 |-  ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) )
39 6 35 37 38 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) )
40 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ P e. A ) -> ( R .\/ P ) e. ( Base ` K ) )
41 5 19 12 40 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( R .\/ P ) e. ( Base ` K ) )
42 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ U e. A /\ S e. A ) -> ( U .\/ S ) e. ( Base ` K ) )
43 5 26 13 42 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( U .\/ S ) e. ( Base ` K ) )
44 9 3 latmcl 
 |-  ( ( K e. Lat /\ ( R .\/ P ) e. ( Base ` K ) /\ ( U .\/ S ) e. ( Base ` K ) ) -> ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) )
45 6 41 43 44 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) )
46 9 2 latjcom 
 |-  ( ( K e. Lat /\ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) /\ ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) ) -> ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) = ( ( ( R .\/ P ) ./\ ( U .\/ S ) ) .\/ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) ) )
47 6 39 45 46 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) = ( ( ( R .\/ P ) ./\ ( U .\/ S ) ) .\/ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) ) )
48 2 4 hlatjcom 
 |-  ( ( K e. HL /\ U e. A /\ S e. A ) -> ( U .\/ S ) = ( S .\/ U ) )
49 5 26 13 48 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( U .\/ S ) = ( S .\/ U ) )
50 21 49 oveq12d 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( R .\/ P ) ./\ ( U .\/ S ) ) = ( ( P .\/ R ) ./\ ( S .\/ U ) ) )
51 2 4 hlatjcom 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) = ( R .\/ Q ) )
52 5 7 19 51 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( Q .\/ R ) = ( R .\/ Q ) )
53 2 4 hlatjcom 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) = ( U .\/ T ) )
54 5 8 26 53 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( T .\/ U ) = ( U .\/ T ) )
55 52 54 oveq12d 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) = ( ( R .\/ Q ) ./\ ( U .\/ T ) ) )
56 50 55 oveq12d 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( R .\/ P ) ./\ ( U .\/ S ) ) .\/ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) ) = ( ( ( P .\/ R ) ./\ ( S .\/ U ) ) .\/ ( ( R .\/ Q ) ./\ ( U .\/ T ) ) ) )
57 47 56 eqtrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) = ( ( ( P .\/ R ) ./\ ( S .\/ U ) ) .\/ ( ( R .\/ Q ) ./\ ( U .\/ T ) ) ) )
58 28 33 57 3brtr4d 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ P ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )